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Quantitative Marketing and Economics

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19.08.2017

The timing of version releases: A dynamic duopoly model

verfasst von: Ron N. Borkovsky

Erschienen in: Quantitative Marketing and Economics | Ausgabe 3/2017

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Abstract

In many R&D-intensive consumer product categories, firms deliver value to consumers through the quality enhancements provided by new and improved versions of existing products. Therefore, important marketing decisions relate to a firm’s strategy for developing quality enhancements and releasing new versions. This paper explores this type of product development using a dynamic duopoly model that endogenizes each firm’s decisions over how much to invest in R&D and when to release new versions. Specifically, I explore how two key industry fundamentals—the degree of horizontal differentiation and the cost of releasing a new version—affect firms’ product development strategies and, accordingly, the evolution of industry structure. I find that varying the degree of horizontal differentiation gives rise to three distinctly different types of competitive dynamics: preemption races when the degree of horizontal differentiation is low; phases of accommodation when it is moderate; and asymmetric R&D wars when it is high. Furthermore, I find that an increase in the cost of releasing a new version can induce firms to compete more aggressively for the lead and, in doing so, release new versions more frequently despite the higher cost.

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P&G (Crest) and Colgate-Palmolive (Colgate) have invested heavily in developing innovations relating to the active ingredient (which fights tooth decay), other therapeutic benefits (e.g., tartar prevention), and packaging, and have introduced these innovations via periodic version releases (McCoy 2001; Parry 2001; Dyer et al. 2004).

Goettler & Gordon (2011) incorporate product durability into a quality ladder model in the Ericson & Pakes (1995) framework and use it to study competition between Intel and AMD. While their model endogenizes firms’ R&D spending decisions, it does not endogenize the timing of version releases; rather, it assumes that a firm releases a new product after each successful innovation.

See Scherer (1967), Kamien & Schwartz (1972), Wilson & Norton (1989), Moorthy & Png (1992), and Bhaskaran & Ramachandran (2011).

Hitsch (2006) devises and estimates a model of a firm’s decisions on whether to launch—and subsequently whether to scrap—new products in the face of demand uncertainty that resolves itself over time (post-launch) as sales are realized. The model differs from the one in this paper in that it (i) focuses on new products with unchanging quality and therefore incorporates neither R&D investment nor repeated releases (but does incorporate advertising); and (ii) does not incorporate strategic interaction between firms.

Aoki (1991), Harris (1991), Budd et al. (1993), and Hörner (2004) explore R&D competition using analytically tractable dynamic stochastic games. To achieve analytic tractability, each assumes that there is a one-dimensional state space that reflects the size of one firm’s lead. While I too restrict attention to the size of a firm’s lead in the product market, my research question cannot be explored using an analytically tractable model because I require two additional states that track the firms’ respective R&D stocks.

Ofek & Sarvary (2003) devise an analytic model (with a one-dimensional state indicating whether a firm is leader or follower) to study dynamic competition in markets in which firms invest in R&D to develop next-generation products. They explore the implications of different advantages that a leader might possess in terms of innovative ability, reputation, and advertising effectiveness. This paper differs from Ofek & Sarvary (2003) in that it focuses on the strategic role of R&D stockpiling and (endogenous) version releases.

This approach is common in the theoretical literature on dynamic quality ladder models; see the papers cited in footnote 5.

I assume that the cost of releasing a new version is private and random because it guarantees the existence of a Markov perfect equilibrium in pure strategies (Doraszelski & Satterthwaite 2010).

In the Appendix, I explain that the Logit demand model can be reinterpreted as an address model (Anderson et al. 1992). I also show that if one replaces the Logit demand model with the Hotelling (1929) address model, the equilibria that the dynamic model admits are qualitatively similar.

The assumption that there is no outside option has two noteworthy implications. First, the model admits an equilibrium in which each firm sets an infinite price and sells to half the market. I rule this out by assuming that prices are finite. Second, the Caplin & Nalebuff (1991) proof of existence and uniqueness does not apply. However, I have succeeded in computing a Nash equilibrium for every value of ω ^m at every parameterization explored in this paper. Moreover, I have not encountered multiple equilibria.

I have assumed that if a firm fails to incorporate a unit of R&D stock into a new version, then that unit of R&D stock is lost. The reasoning and the examples that I provide in Section 2.2 suggest that this assumption is reasonable for CPG and consumer electronics categories, among others. Alternatively, one could assume that if a firm fails to incorporate a unit of R&D stock, it retains it and can try again in the future. This entails assuming that when a firm releases a new version, its R&D stock transitions from \({\omega ^{R}_{i}}\) to \({\omega ^{R}_{i}} - \bar {\omega }^{R}_{i} \) instead of zero; the rest of the model is unchanged. I thank Referee 1 for this suggestion.

See the papers cited in footnotes 5 and 6.

In Section 3, I set L _m = 20 and L _R = 10. It follows that the number of industry states is 41 × 11² = 4961. However, had I not reduced the dimensionality of the state space (from four to three), then there would have been 21² × 11² = 53, 361 industry states.

First, some uncertainty exists irrespective of how much test marketing a firm does. Furthermore, CPG firms often elect to do relatively little test marketing because it is “slow, expensive, and open to spying and sabotage” (Baker et al. 2000). Second, the presence of such uncertainty relates to the idea that firms sometimes inadvertently focus on improving products from a technical standpoint—instead of focusing on satisfying customers’ needs and wants—and therefore may make improvements that customers do not value (Levitt 1960). Finally, although Coke is not characterized by frequent version releases, the infamous 1985 release of a new version of Coke—which replaced the previous version—provides an excellent example of this phenomenon. The new version of Coke failed because of tremendous public backlash despite much market research suggesting that it would be a success (Prendergrast 2000).

For example, Pakes & McGuire (1994), Gowrisankaran & Town (1997), and Borkovsky et al. (2012) assume that a consumer’s utility is characterized by diminishing returns to quality that set in very quickly beyond some quality threshold. I am unable to take this particular approach because in order to reduce the dimensionality of the state space, I have assumed that a consumer’s utility is linear in quality.

In the Appendix, I show that a version of the model with a very small state space is not rich enough to admit the results presented in Section 6.

Additionally, allowing for a wide range of possible release costs yields equilibria with a wide range of release probabilities (across industry states), which makes it easier to discern the different equilibrium behaviors that arise. I have verified that qualitatively similar behaviors arise for narrower ranges of release costs. However, narrower ranges of release costs tend to give rise to greater convergence problems for the algorithm described below. Finally, Section 6.2 explores the effects of changes in the range of release costs by computing equilibria for higher values of G _l and G _u.

Because firms are symmetric, firm 2’s equilibrium price and profit functions are symmetric to firm 1’s, i.e., p ₂(ω _m) = p ₁(−ω _m) and π ₂(ω _m) = π ₁(−ω _m).

In this respect, the simplified model is similar to earlier quality ladder models (e.g., Pakes & McGuire 1994, Borkovsky et al. 2012), the only difference being that, as explained in Section 2.1, I reduce the dimensionality of the state space.

One derives the simplified model from the model in Section 2 by setting the release cost to zero (G _l = G _u = 0) and the probability that a firm successfully incorporates its entire R&D stock (when releasing a new version) to one—i.e., \(s\left ({\omega ^{R}_{i}} | {\omega ^{R}_{i}}\right )=1\).

Because firms are symmetric, firm 2’s R&D investment policy function is symmetric to firm 1’s, i.e., x ₂(ω ^m) = x ₁(−ω ^m).

I also compute equilibria for fine discretization of a wide range of σ values and present the equilibrium correspondence in the Appendix.

In this case, the matrices graphed in the bottom row are transposes of the corresponding matrices in the top row; this is because I restrict attention to symmetric equilibria and because firms are tied in the product market. However, this will not be the case for cross-sections for which ω ^m ≠ 0.

The bottom panels in Fig. 3 show that there is a roughly triangular region in the \(({\omega ^{R}_{1}},{\omega ^{R}_{2}})\) grid—where \({\omega ^{R}_{1}}\) is sufficiently high and \({\omega ^{R}_{2}}\) is sufficiently low—in which firm 2 gives up. The upper panels show that there is an analogous region in which firm 1 gives up.

When one firm gains a lead of 20 in the product market, both firms neither invest nor release new versions, i.e., the industry reaches an absorbing state. It does however take a very long time until this occurs—approximately 60 periods, which is twice the industry’s expected lifespan. The standard approach to addressing this issue in Ericson & Pakes (1995) models is to assume that a firm experiences decreasing returns as it approaches the upper edge of the state space. I have verified that a version of the model in which the effectiveness of investment, α, decreases in the size of a firm’s lead yields equilibria that are qualitatively similar to those presented in Section 6, the only difference being that firms never stop investing or releasing new versions and accordingly neither firm ever achieves a maximal product market lead of 20. These equilibria are available upon request.

For example, in industry state (0, 6, 2)—in which firm 2 gives up—firm 1 is the likely product market leader not only because of its R&D stock advantage, but also because it updates with much higher probability (0.4272) than firm 2 (0.0395).

A version release has a direct effect and a strategic effect. The direct effect is that a version release enhances a firm’s product quality and accordingly its profits. The larger a firm’s R&D stock, the more it enhances the firm’s (expected) profits when the firm releases a new version, and accordingly the higher the firm’s release probability should be. Hence, if one considers only the direct effect, one would expect a firm’s release probability to be strictly increasing in its R&D stock, as is the case in a monopolistic version of the model. The fact that a firm’s release probability is not strictly increasing in its R&D stock can be attributed to the strategic effect, i.e., the impact of a version release on a rival firm’s behavior.

Some preemption in terms of version releases still remains because of the uncertain nature of the release cost; see the Appendix for details.

In multistage patent races, if R&D investment has a deterministic impact on innovation (Fudenberg et al. 1983; Harris & Vickers 1985; Lippman & McCardle 1988), then ε -preemption arises, i.e., once a firm gains an arbitrarily small lead, it induces its rival to immediately drop out of the race. If R&D investment has a stochastic impact on innovation (Grossman & Shapiro 1987; Harris & Vickers 1987; Lippman & McCardle 1987), then the leader invests more than the follower and is therefore likely to expand its lead until it ultimately induces its rival to drop out.

Similarly, because this is a symmetric equilibrium and in the ω ^m = 0 cross-section, firms are tied in the product market, there are identical trenches along \({\omega _{1}^{R}}=1\) for \({\omega _{2}^{R}}\geq 3\) and along \({\omega _{1}^{R}}=4\) for \({\omega _{2}^{R}}\geq 8\), in which the roles of the firms are reversed.

I have found three additional equilibria for the σ = 2 parameterization that are qualitatively similar to the one presented in Fig. 5, the only difference being that the accommodative trenches arise at different R&D stock levels; see the Appendix for details.

The probability that firm 2 is the product market leader (ω _m < 0) 100 periods after the industry enters state ω = (0, 3, 1) is 1.88%.

In fact, for ω1R ≥ 5, firm 2 does not invest at all while in this trench. Therefore, its R&D stock remains fixed at \({\omega _{2}^{R}}=1\) as long as long as neither firm releases a new version.

I plot the ω ^m = 0 cross-sections of only firm 1’s policy functions; as in Figs. 3 and 5, the cross-sections of firm 2’s policy functions are simply the transposes of those presented for firm 1.

The panel in the middle row and middle column of Fig. 7 shows that when a firm gains a large R&D stock lead, it updates with high probability. Hence, a follower invests heavily in R&D in hopes of preventing the leader from gaining such a large lead. Because the follower updates with low probability, the leader faces no similar incentive.

In contrast, in earlier quality ladder models (e.g., Borkovsky et al. 2012) and more generally in dynamic models of industry evolution (e.g., Besanko et al. 2010), symmetric industry structure tends to arise when firms compete less aggressively.

The equilibria in the middle and lower panels of Fig. 7 are also characterized by accommodation. While the ω ^m = 0 cross sections in the lower panels do not include any accommodative trenches and the ones in the middle panels include only one (at \({\omega _{2}^{R}}=0\) for \({\omega _{1}^{R}}\geq 9\)), there are several prominent accommodative trenches in other cross-sections for both equilibria.

The bottom panels of Fig. 2 show that in the benchmark model, when the degree of horizontal differentiation is high, as the laggard falls behind it simply reduces its R&D investment and the leader increases its R&D investment (for ω ^m ≤ 10), making it extremely unlikely that the laggard narrows the leader’s lead; accordingly, an asymmetric industry structure arises.

Besanko et al. (2010) explain why their model of dynamic price competition admits multiple equilibria and, specifically, how different equilibria are underpinned by different beliefs about the future (see pp. 492–493). The same explanation applies to my model.

The relatively small non-monotonicities in the left panel of Fig. 8 arise because of slight qualitative changes in the equilibrium policy functions.

Indirect network effects existed because web content developers preferred to produce content for browsers with large user bases and users preferred browsers for which much compatible content was available (Bresnahan 2001).

Nair et al. (2004), Markovich (2008), Markovich & Moenius (2009), Dubé et al. (2010), and Lee (2013) explore the impact of indirect network effects in an oligopolistic context.

To incorporate product durability into the model, in addition to including forward-looking consumers and dynamic pricing, one would augment the state space to include the ownership distribution across products (Goettler & Gordon 2011).

Incorporating entry and exit into the model would be theoretically straightforward. However, it would increase computational burden because one would not be able to restrict attention to the difference between firms’ respective product qualities. If one firm exits, the remaining incumbent firm would have to be characterized by its product’s absolute quality. Therefore, it would be necessary to retain absolute as opposed to relative product qualities in the state space.

The notion of learning about uncertain demand from observed sales is explored in Hitsch (2006).

In the diaper category, several innovations have been introduced via product line extensions, e.g., gel technology (Pampers 1986), Velcro tabs (Huggies 1993, Pampers 1994), and stretchable side panels (Pampers 1994) (Parry & Jones 2001). These innovations were later broadly incorporated into the firms’ respective product lines. Both P&G and Kimberly-Clark have also introduced many innovations directly into new versions of existing products, e.g, tape closures (Pampers 1971), stay-dry lining (Pampers 1976), a waist shield (Pampers 1985), and contoured elastic leg bands (Huggies 1992) (Dyer et al. 2004; Parry & Jones 2001).

The literature on analytically tractable quality ladder models (see footnote 5) assumes that an exogenous profit function is (weakly) increasing in the size of a firm’s lead. In the empirical (Gowrisankaran & Town 1997; Goettler & Gordon 2011; Borkovsky et al. 2017) and computational applied theory (Pakes & McGuire 1994; Markovich 2008; Markovich & Moenius 2009; Borkovsky et al. 2012; Goettler & Gordon 2014) literatures, the (endogenous) static profit function is strictly increasing in a firm’s quality because—as in my model—the marginal utility of quality is constant across consumers.

The literature on endogenous quality choice explores the decisions that firms face in positioning new products relative to their competitors (see p. 141 of Moorthy1988). The literature on quality ladder models explores competition amongst firms that repeatedly invest in R&D to incrementally enhance (or sustain) the quality of existing products while also repeatedly engaging in product market competition.

This insight stems from Referee 2’s comments, for which I am extremely grateful.

The version release decision in this model is analogous to the entry and exit decisions in Doraszelski & Satterthwaite (2010); by assuming that the release cost is random and privately known, I “purify” the mixed strategy equilibria that the model would admit were the release cost fixed (Harsanyi 1973).

Because I restrict attention to symmetric equilibria, all relevant differences between firms are reflected in the industry state ω. Accordingly each firm’s behavior is a function of only the size of its product market advantage/disadvantage and the status of the R&D stock race. For example, firm 1’s behavior in industry state (7,4,2) is identical to firm 2’s behavior in industry state (-7,2,4); in each case, the firm in question has a product market lead of 7 and has an R&D stock lead of 4 to 2.

For this specification, I have found that setting ρ = 0.95 mitigates the effect of the edge of the state space. Alternatively, one could assume s(⋅|ω i R) is binomial, i.e., each unit of R&D stock is successfully incorporated with the same probability (q _n = ρ for all n). However, if one uses the binomial specification, then one must assume a much lower success probability in order to successfully mitigate the effects of the edge of the state space. This lower success probability weakens investment incentives on the interior of the state space much more than the generalized binomial specification described above. In this respect, the generalized binomial specification is similar in spirit to the approach taken in Pakes & McGuire (1994), Gowrisankaran & Town (1997), and Borkovsky et al. (2012); see footnote 15.

Reframing the timing of the game does not change the model. Because the model is infinitely repeated, one can define the value function and accordingly derive the Bellman equation from the perspective of any step in the timing of a period—as long as one applies the discount factor β at the beginning of the actual period. Therefore, when deriving the Bellman equation, I apply the discount factor β at the beginning of subperiod 1.

While I have succeeded in computing at least one equilibrium for each parameterization, there is no guarantee that all equilibria have been found.

This simply entails setting \(s(\omega _{i}^{R}|\omega _{i}^{R})=1\) for all \(\omega _{i}^{R}\in \Omega ^{R}\).

This raises the question of why the release probabilities in the top row of Fig. 14 are characterized by any preemption whatsoever. The reason is that there is also uncertainty over the cost of releasing a new version. (Recall that this uncertainty is required to guarantee the existence of an equilibrium in pure strategies.) Hence, a firm that releases a new version can gain a strategic advantage over a rival because to catch up the rival would need to draw a release cost that is low enough to make updating optimal.

Recall that the ω = 0 cross-sections of firm 2’s policy functions are the transposes of those presented for firm 1.

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Titel: The timing of version releases: A dynamic duopoly model
verfasst von: Ron N. Borkovsky
Publikationsdatum: 19.08.2017
Verlag: Springer US
Erschienen in: Quantitative Marketing and Economics / Ausgabe 3/2017
Print ISSN: 1570-7156
Elektronische ISSN: 1573-711X
DOI: https://doi.org/10.1007/s11129-017-9186-9